Characterization of the accuracy of the fast multipole method in particle simulations

Abstract

SUMMARYThe fast multipole method (FMM) is a fast summation algorithm capable of accelerating pairwise interaction calculations, known as N -body problems, from an algorithmic complexity of O(N 2 ) to O(N ) for N particles. The algorithm has brought a dramatic increase in the capability of particle simulations in many application areas, such as electrostatics, particle formulations of fluid mechanics, and others. Although the literature on the subject provides theoretical error bounds for the FMM approximation, there are not many reports on the measured errors in a suite of computational experiments that characterize the accuracy of the method in relation with the different parameters available to the user. We have performed such an experimental investigation, and summarized the results of about 1500 calculations using the FMM algorithm, applied to the 2D vortex particle method. In addition to the more standard diagnostic of the maximum error, we supply illustrations of the spatial distribution of the errors, offering visual evidence of all the contributing factors to the overall approximation accuracy: multipole expansion, local expansion, hierarchical spatial decomposition (interaction lists, local domain, far domain). This presentation is a contribution to any researcher wishing to incorporate the FMM acceleration to their application code, as it aids in understanding where accuracy is gained or compromised.

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier Series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions

“…In spite of its great cost, this option is widely used with radial basis functions (RBF) because of its simplicity. A much faster procedure is to use a so-called Fast Summation to sum RBF series and a preconditioned iteration, repeatedly calling a Fast Summation routine, for interpolation [98,26]. Because these routines are complicated, they are not yet widely used with RBFs, so Table 7 lists the costs for both direct and iterative RBF interpolation.…”

A fast resurrected core-spreading vortex method with no-slip boundary conditions

“…When P > P cr , this error dominates over the truncation error. A detailed investigation of all factors contributing to the overall accuracy of the approximation associated with the fast multipole method is found in [37]. The nearby blobs are grouped on taking advantage both of the square cells generated in the previous section for merging similar and nearby blobs and of the concept of adaptive domain decomposition that Carrier et al proposed [24]: a minimum square domain is identified that must cover the entire flow field (all existing vortex blobs and objectives) and that comprises square cells of size gr max .…”

Parallel Implementation of Vortex Element Method on CPUs and GPUs

“…Nevertheless, considerable improvement in time of computation can be obtained due to approximate fast multipole method usage for vortex-to-vortex influences calculation. It reduces computational cost from O(N 2 ) to O(N log N ) [14,15]. Fast multipole method is based on a binary tree constructing, where its root contains all the vortex elements, cells of the first level contain vortex elements which are placed in the halves of the root cell, etc.…”